Universal scaling laws for charge-carrier interactions with quantum confinement in lead-halide perovskites

Lead halide perovskites open great prospects for optoelectronics and a wealth of potential applications in quantum optical and spin-based technologies. Precise knowledge of the fundamental optical and spin properties of charge-carrier complexes at the origin of their luminescence is crucial in view of the development of these applications. On nearly bulk Cesium-Lead-Bromide single perovskite nanocrystals, which are the test bench materials for next-generation devices as well as theoretical modeling, we perform low temperature magneto-optical spectroscopy to reveal their entire band-edge exciton fine structure and charge-complex binding energies. We demonstrate that the ground exciton state is dark and lays several millielectronvolts below the lowest bright exciton sublevels, which settles the debate on the bright-dark exciton level ordering in these materials. More importantly, combining these results with spectroscopic measurements on various perovskite nanocrystal compounds, we show evidence for universal scaling laws relating the exciton fine structure splitting, the trion and biexciton binding energies to the band-edge exciton energy in lead-halide perovskite nanostructures, regardless of their chemical composition. These scaling laws solely based on quantum confinement effects and dimensionless energies offer a general predictive picture for the interaction energies within charge-carrier complexes photo-generated in these emerging semiconductor nanostructures.

The PL intensity autocorrelation functions of 8 single NCs measured at 3.5 K and in zero field are assorted in ascending order of zero-delay autocorrelation ! 0 in a, c, e, g, i, k, m, o. The values of ! 0 range from 2.5 up to 32. Their corresponding autocorrelation functions at 7 T are presented in b, d, f, h, j, l, n, p with the same vertical scales, showing evidence for a reduction of photon bunching effect under magnetic field. These behaviors have been observed for all nearly-bulk CsPbBr 3 NCs. Note that the slight asymmetry of the autocorrelation histograms m and o may be due to a residual cross-talk effect between the two avalanche photodiodes.
As explained and modeled in Ref. 1 , photon bunching takes its origin in the fact that the long-lived ground state stores the exciton for its long lifetime, which promotes the generation of biexcitons in the NC, followed by biexciton to zero-exciton two-photon cascades. The weakening of photon bunching with the application of magnetic fields results from the shortening of the dark state lifetime. p" 0"T" 0"T" 0"T" 0"T" 0"T" 0"T" 0"T" 0"T" 7"T" 7"T" 7"T" 7"T" 7"T" 7"T" 7"T" 7"T" Supplementary Fig. 8: PL decay and photon coincidence histograms under magnetic fields. a, Correlated PL decays and photon coincidence histograms for the same single NC at 0 T and at 7 T, the temperature being 3.5 K. This NC is the one whose PL spectra are displayed in Supplementary Fig. 6b. The correlations are well reproduced in b, c with the model developed in Ref. 1 , taking ! = 10 ns -1 , ! = 0.1 ns -1 , = 0.025 ns -1 , !! = 0.12, ! = 0.01 ns -1 at 0 T (b) and ! = 0.125 ns -1 at 7 T (c) (i.e. a lifetime of 8 ns given by the long component of the PL decay in a). %(τ)% τ%(ns)% g (2) %(τ)% τ%(ns)% a" b" c" Supplementary Fig. 9: Spectral trail of a NC at 7 T under high excitation intensity These series of consecutive PL spectra were recorded at 50 kW cm -2 on the same NC as in Fig. 3a. One can notice that the trion binding energies experience a much larger spread than the biexciton ones in Fig. 4. Since the NCs are either in the neutral or in the charged state, the exciton and trion spectral features are inherently measured on different emission spectra. Spectral jumps occurring during these emission switches are frequent and may lead to dispersion in the measured trion binding energies. Charge complexes such as trions are indeed highly sensitive to the local dielectric environment and to charge redistributions located at the surface of the NC. Time"(s)" Energy"(eV)" Triplet" Singlet" Supplementary Fig. 10: Dependence of the PL spectra on the excitation intensity. a, Evolution of the spectral features assigned to the exciton triplet (X) and the biexciton-toexciton triplet (XX) under increasing excitation intensities ranging from I = 0.8 kW cm -2 to 4.1 I. b, The ratio of the integrated intensities of the XX and X spectral structures is plotted as a function of the integrated intensities of the X triplet and shows a linear behavior, as a distinctive signature of XX biexciton emission.
x"10" XX" X" a" b" Supplementary Fig. 11: Distribution of the biexciton binding energy in CsPbBr 3 large NCs. a, Low-temperature PL spectrum of a single NC at high excitation intensity (50 kW cm -2 ) and 7 T. The four lines at energies higher than 2.32 eV are assigned to recombination ZPLs of the triplet (named X 1 , X 2 , X 3 ) and the dark ground singlet. The red-shifted lines named XX 1 , XX 2 , XX 3 are assigned to the biexciton-to-exciton transitions, as shown in the inset. b, Distribution of the biexciton binding energy, measured as the energy offset of the XX 2 line with respect to the X 2 line, as a function of the exciton recombination energy (taken at the X 2 line).
G" Supplementary Fig. 12: Polarization of the exciton and biexciton recombination lines. a,b Polarized PL spectra of a single CsPb(Cl x Br 1-x ) 3 single NC with x~0.3 and size ~30 nm (See Methods and Supplementary Fig. 13), for various analyzer angles ranging from 0° to 180°. The series of spectra in a are zoomed on the exciton line triplet (X 1 , X 2 , X 3 ZPLs), while those in b are zoomed on the biexciton recombination ZPLs (named XX 1 , XX 2 , XX 3 ). The evolutions of the exciton and biexciton ZPL-intensities as a function of the polarizer angle are displayed in the polar plots c and d, respectively. The polar plot in c shows evidence that the transition dipoles associated to X 1 and X 3 ZPLs have nearly linear and orthogonal polarizations in the focal plane, while the weak X 2 line with a weak polarization character is compatible with a transition dipole oriented along the optical axis. The striking similarity between the polar plots provides further evidence of the correspondence between the biexciton-to-exciton and exciton-to-zero-exciton transitions, as a result of symmetry (See Supplementary Note 2).   The effective mass and the effective dielectric constant were derived from the hydrogen model combined with either (i) the approximation = ! !"# 4 ! , using estimations of the Kane momentum 2,3 or (ii) the dependence of the exciton diamagnetic-shift coefficient on and ε !"" 4 . For the mixed halide compound CsPb(Cl x Br 1-x ) 3 with x~0.24, the lowtemperature band gap is estimated using a linear Vegard's interpolation law provide estimates for the reduced mass and effective dielectric constant.
Supplementary Note 1

Smearing of the electronic and vibrational density of states in bulk perovskites
We address the possible origins for the variation of emission energies among different single CsPbBr 3 NCs, especially down to the bulk limit (Fig. 2). Recent experimental investigations on bulk 3D perovskites combining various techniques looking at several orders of magnitude for the dynamics (Neutron, Raman and Brillouin scattering, NMR, dielectric and ultrasonic measurements) have indeed shown that besides soft acoustic and polar damped vibrations, ultraslow structural relaxation exists in these bulk materials. Its microscopic origins may be stochastic deviations of cation positions or torsions of the halogen octahedra, which translate into a complex potential landscape felt by the exciton. This is a consequence of the very strong lattice anharmonicity or intrinsic polymorphous nature of the perovskite structure [5][6][7] . A first conventional theoretical approach consists in performing ab-initio molecular dynamics (AIMD). Quarti et al. 8 reported early in 2015 strong stochastic and localized band gap fluctuations over time for 3D bulk halide perovskites using AIMD. It can be further refined in the constant-temperature, constant-pressure ensemble (NPT) to capture the coupling of disorder to acoustic-like fluctuations 9 . These band gap fluctuations are reminiscent of the spectral diffusion of the emission. However, AIMD's calculations have strong limitations. The computational cost is especially prohibitive, with various limitations on the total computational time and hence the quality of the structural dynamics sampling, the size of the simulation super-cell, the lattice temperature (high temperature are very often used to artificially speed up the simulations) and the level of theory used, especially to extract accurate information on the physics of the electron-phonon coupling.
In order to get a broader view on the general consequences of these fluctuations, a new theoretical framework is proposed to extract the smearing of the vibrational and electronic density of states, which leads to a broadening of the distribution of the emission energies. In the Supplementary Fig. 14, we present the effect of structural disorder on the phonons and electronic structure of CsPbBr 3 . Supplementary Fig. 14a compares the phonon spectral function (color map) and phonon dispersion (black lines) calculated using the polymorphous (ground state structural disorder) and ideal monomorphous (nuclei fixed at their highsymmetry positions) cubic networks, respectively. The polymorphous description was shown recently to give a much better description of XRD pair distribution functions. Interestingly, accounting for static disorder in CsPbBr 3 leads to a large smearing of the optical phonons, displaying an over-damped anharmonic behavior across the reciprocal lattice similar to the one observed experimentally by various phonon spectroscopy techniques. Supplementary Fig.  14b compares the associated electron spectral function (color map) and band structure (black curves) calculated using the polymorphous and monomorphous networks, respectively, including the effect of spin-orbit coupling. For the calculation of the electron spectral function, we have included the effect of electron-phonon coupling at 0 K using the special displacement method 10,11 . Note that the electronic band-gap between black curves in underestimated at the theoretical level of DFT. Nevertheless, apart from a remarkable bandgap increase arising from both structural disorder and electron-phonon coupling, our computations reveal a considerable smearing of the electron states, which allows various optical transitions. To get a more quantitative agreement for the effective band gap, this new methodology could be improved by relying on other functionals, but the main physical conclusion concerning the smearing of both vibrational and electronic densities of states will not be changed.
It is well established that in hybrid perovskite crystals, the various orientations of the electric dipole of the organic molecular cation play an important role in creating structural electrostatic potential fluctuation. Such fluctuations are also present in inorganic perovskites (without polar molecules) as a result of stochastic deviations of cation positions or torsions of the halogen octahedra (Ref. 12 and references therein). The wavefunction of the exciton centerof-mass evolves in the resulting potential landscape and may be spatially localized in its minima at low temperatures. Such potential wells confining the exciton should have a typical size larger than the exciton Bohr radius and thus be in little number in the large CsPbBr 3 NCs of this study. This picture is supported by the experimental observation of exciton diffusion towards low energy localized states in 2D perovskites 13,14 . Furthermore, exciton localization towards the lowest-energy potential wells was found to occur at low temperature 14 . Supplementary Fig. 14: Calculated phonon and electron spectral functions. a, Phonon spectral function of cubic polymorphous (disordered) CsPbBr 3 (color map) for frequencies up to 11 meV, plotted along the M-Γ-R high-symmetry path of the reciprocal lattice space. b, Electron spectral function at 0 K of cubic polymorphous CsPbBr 3 (color map) plotted along the X-R-M high-symmetry path of the reciprocal lattice space. Black lines represent the phonon dispersion (a) and electron band structure (b), calculated using the monomorphous cubic 'ideal' and undistorted network, i.e. without accounting for structural disorder. Ab-initio calculations were performed within the density functional theory (DFT), using plane waves as basis functions in the PBEsol approximation 15 , as implemented in the electronic structure package Quantum Espresso 16 . We also used optimized norm-conserving Vanderbilt pseudopotentials as proposed by Haman 17 . We started our calculations from the monomorphous structure, using the unit cell of cubic CsPbBr 3 (5 atoms) with the nuclei fixed at their high symmetry positions (space group Pm3m). The lattice constant was fixed to the DFT-PBEsol relaxed value of 5.874 Å and the plane wave energy cutoff was set to 120 R y . To obtain the polymorphous structures 6 of cubic CsPbBr 3 we relax the system to its disordered ground state in a 2×2×2 supercell. To this aim, we initially displaced the nuclei away from their high-symmetry positions using special displacements 10,11 generated for T = 0 K, and allow the system to relax until the residual force component per atom was less than 3×10 -4 eV/ Å. The sampling of the Brillouin zone of the unit cell and 2×2×2 supercell was performed using uniform 6×6×6 and 3×3×3 k-grids, respectively. The phonons of the monomorphous and polymorphous structures were obtained by means of the frozen-phonon method 18 and Fourier interpolation of the dynamical matrices. To account for electron-phonon coupling effects on the electronic structure of the polymorphous network, we used the special displacement method 10,11 , enforcing a smooth Berry connection between the phonon eigenmodes as implemented in the ZG branch of the EPW code 19 . Phonon and electron spectral functions of the polymorphous structure, shown as color maps, were evaluated using the band structure unfolding technique as described in Ref. 10 .

Exciton fine structure and exciton dark-bright splitting
In the case of the dark-bright splitting of the exciton, a specific dimensional analysis is needed. As shown in previous studies conducted on FAPbBr 3 20 and CsPbI 3 1 perovskites, the dark-bright splitting !" is essentially set by the long-range component of the electron-hole exchange interaction, which is directly related to the dimensionless ratio !"" ! and to the difference between the energies of longitudinal and transverse excitonic polaritons ℏ !" 3, where !" being the Kane interband momentum matrix element and ! the electron mass 21 . Indeed, since the seminal paper of Onodera and Toyozawa 22 , it is recognized that the problems of exchange interaction and transverse and longitudinal excitons are connected 23 . Thus, ℏ !" shall afford a more adequate normalization factor of the dark-bright splitting as compared to the exciton Rydberg energy ! . This quantity can be related to ! and !"# considering the Kane energy !"#$ = !" ! 2 ! introduced as a fundamental constant of 2D 24 and 3D 25 perovskites. It was experimentally shown that a single Kane energy is enough to describe the evolution of the effective masses with bandgaps for various compositions of 3D bulk metal halide perovskites 2,3 . Using !"#$ !"# ∝ !! leads to the proportionality coefficients being material independent. Therefore, using the dimensionless quantity !" ! ! !"# should lead to a universal dependence of the exciton dark-bright splitting as a function of the quantum confinement. This is demonstrated in Fig.  4c, where a clear correlation between the dimensionless bright-dark splitting and the exciton recombination energy, after bandgap subtraction, is evidenced.
We now justify why a scaling factor designed to account for the long-range exchange interaction remains relevant for an experimental quantity related to the sum of the short-range and long-range contributions. We have numerically inspected the bulk limits of both contributions for various perovskites. Rather than considering a few values spread in the literature 26 for the short-range component, we use instead the general bulk formula proposed by Ben Aich et al. 27 and derived from Rössler et al. 28 .
Indeed, this formula is proposed in the same spirit as the present work and aims at proposing a law based on an empirical factor C=107.6 meV.nm 3 . This contribution is computed in the Supplementary Table 2, together with the bulk limit of the long range part Δ !",!",!"#$ = ℏ! !" ! , for all compounds relevant to perovskite nanostructures. Interestingly, when the proper scaling factor ! ! !"# is introduced, all these perovskites yield very similar dimensionless bulk limits of the exchange energy in the 5-6 range, in reasonable agreement with the bulklimit extrapolation of the experimental master curve (Fig. 4c). Since the short-range and longrange contributions of the exchange interaction are predicted to have the same NC sizedependence in the strongly confined regime 21 , the applicability of the scaling factor ! ! !"# over the whole range of confinement regimes is strengthened. Supplementary Table 2: Contributions of the short-range and long-range electron-hole exchange interactions, using the normalization factor .

Correlated polarizations for the biexciton and exciton emissions
This section aims at providing the explanations for the correlation between polarizations of the biexciton-to-exciton and exciton-to-ground-state transition lines ( Supplementary Fig. 12). As already discussed in Ref. 29 , the double group irreducible representations of the D 2h point group are suitable for the orthorhombic bulk phase of CsPbBr 3 and lead to the following decomposition of the exciton fine structure: For biexciton states related to the same bulk Bloch states, one obtains: The ! irreducible representation is the only one related to a total momentum equal to zero corresponding to a complete filling of the doubly degenerated conduction band edge, and thus to the biexciton ground state. A transition from the ! biexciton ground state to one of the three bright exciton states as described by !"#$% !.!. !"# corresponds to an optical transition with a polarization along one of the three , , crystallographic axes, since the vectorial representation relevant for the dipolar electric Hamiltonian !.!. is = !" ⊕ !" ⊕ !" . This transition is therefore followed by an optical transition from an exciton state to the ground state